Pass the Pigs TM Statistics 104 - Laboratory 7 On last weeks lab we looked at probabilities associated with outcomes of the game Pass the Pigs TM. This week we will look at random variables associated with this game. 1. Scoring Points Various combinations of how the pigs land score points or result in you losing points for that round or the game. Below are the points that can be scored on any roll of a pair of pig dice and the associated probabilities. The probabilities are derived from the 6,000 rolls of the pig dice presented in last week s lab. a) Explain why this is a probability distribution. b) Draw a histogram for the distribution and describe the shape of the distribution. c) What is the least likely number of points scored? What is the associated probability? d) What is the probability of scoring 15 or fewer points? e) Find the mean number of points scored on a single roll of two pig dice. 2. What happens on several rolls? Points Probability 0 0.2170 1 0.2173 5 0.3895 10 0.0847 15 0.0285 20 0.0622 25 0.0003 40 0.0003 60 0.0002 Consider rolling the pair of pig dice more than once and noting whether you score some points or score zero points and have to pass the pigs. a) What is the probability of scoring zero points on a single roll of a pair of pig dice? b) Assuming rolls of the pair of pig dice are independent what is the probability of scoring zero points on two consecutive rolls? c) Assuming rolls of the pair of pig dice are independent what is the probability of scoring zero points on three consecutive rolls? d) Assuming rolls of the pair of pig dice are independent what is the probability of scoring points on a single roll? e) Assuming rolls of the pair of pig dice are independent what is the probability of scoring points on two consecutive rolls? f) Assuming rolls of the pair of pig dice are independent what is the probability of scoring points on three consecutive rolls? 1
For 20 rolls of the pair of pig dice, the probability distribution for the number of rolls on which points are scored, x, is given below. For example, the chance that 10 rolls, out of 20, will score points is P(10) = 0.0037. x P(x) x P(x) x P(x) 0 0.0000 7 0.0000 14 0.1318 1 0.0000 8 0.0002 15 0.1902 2 0.0000 9 0.0009 16 0.2144 3 0.0000 10 0.0037 17 0.1821 4 0.0000 11 0.0122 18 0.1095 5 0.0000 12 0.0329 19 0.0416 6 0.0000 13 0.0730 20 0.0075 g) Explain why this is a probability distribution. h) Draw a histogram for the distribution and describe the shape of the distribution. i) What is the probability that you score points on all 20 rolls? j) What is the probability that you score points on 16 or more rolls? k) What is the probability that you score points on 7 or fewer rolls? l) What is the mean of the distribution, i.e. the mean number of rolls on which you score points? m) What is the standard deviation of the distribution? 3. Waiting to Pass the Pigs TM Another random variable that can arise from consecutive rolls of the pigs is to count the number of rolls until you have to pass the pigs, i.e. you score zero points. It is reasonable to assume that consecutive rolls are independent. When thinking about this problem, think about what has to happen on the rolls leading up to you passing the pigs. a) What is the probability that you pass the pigs after just one roll? b) What is the probability that you pass the pigs after the second roll? c) What is the probability that you pass the pigs after the third roll? d) What is the probability that you pass the pigs after the fourth roll? e) What is the probability that you pass the pigs after the fifth roll? f) What pattern do you see in the probabilities? Extra Credit g) Write a general formula for the probability that you pass the pigs after the x th roll. h) The mean of this random variable is 4.608 rolls. Why does this make sense? 2
Statistics 104 - Laboratory 7 Group Answer Sheet Names of Group Members:,, 1. Scoring Points a) Explain why this is a probability distribution. b) Draw a histogram for the distribution and describe the shape of the distribution. c) What is the least likely number of points scored? What is the associated probability? d) What is the probability of scoring 15 or fewer points? e) Find the mean number of points scored on a single roll of two pig dice. 3
2. What happens on several rolls? a) What is the probability of scoring zero points on a single roll of a pair of pig dice? b) Assuming rolls of the pair of pig dice are independent what is the probability of scoring zero points on two consecutive rolls? c) Assuming rolls of the pair of pig dice are independent what is the probability of scoring zero points on three consecutive rolls? d) Assuming rolls of the pair of pig dice are independent what is the probability of scoring points on a single roll of two pig dice? e) Assuming rolls of the pair of pig dice are independent what is the probability of scoring points on two consecutive rolls? f) Assuming rolls of the pair of pig dice are independent what is the probability of scoring points on three consecutive rolls? g) Explain why this is a probability distribution. h) Draw a histogram for the distribution and describe the shape of the distribution. 4
i) What is the probability that you score points on all 20 rolls? j) What is the probability that you score points on 16 or more rolls? k) What is the probability that you score points on 7 or fewer rolls? l) What is the mean of the distribution, i.e. the mean number of rolls on which you score points? m) What is the standard deviation of the distribution? 3. Waiting to Pass the Pigs TM a) What is the probability that you pass the pigs after just one roll? b) What is the probability that you pass the pigs after the second roll? c) What is the probability that you pass the pigs after the third roll? d) What is the probability that you have to pass the pigs after the fourth roll? e) What is the probability that you pass the pigs after the fifth roll? f) What pattern do you see in the probabilities? Extra Credit a) Write a general formula for the probability that you pass the pigs after the x th roll. b) The mean of this random variable is 4.608 rolls. Why does this make sense? 5